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Chapter 5: Problem 15

Factor out the greatest common factor in each expression. $$x^{3}-5 x$$

Short Answer

Expert verified
x(x^2-5)

Step by step solution

01

Identify the terms

The given expression is . Identify the two terms in the expression: .
02

Find the Greatest Common Factor (GCF)

Determine the greatest common factor (GCF) of the terms and . The GCF in this case is .
03

Factor out the GCF

Factor the GCF from each term in the expression -5 . This gives ( - 5).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operators (such as addition, subtraction, multiplication, and division).
These expressions can represent real-world scenarios and can be manipulated using various algebraic rules and techniques.
Understanding algebraic expressions is fundamental to mastering more complex math concepts.

Components of algebraic expressions include:
  • Variables: Symbols (usually letters) that represent unknown values.
  • Constants: Fixed numbers that do not change.
  • Operators: Symbols that show mathematical operations (e.g., +, -, *, /).

For instance, in the expression \(x^{3} - 5x\), \(x\) is a variable, \(3\) and \(-5\) are constants, and \(-\) is an operator.
Breaking down expressions into these components allows us to apply specific techniques, such as factoring or expanding, to simplify or solve them.
Greatest Common Factor
The Greatest Common Factor (GCF) is the largest number (or expression) that divides two or more numbers (or expressions) without leaving a remainder.
Finding the GCF is an essential step in simplifying expressions and solving equations.

To find the GCF of two algebraic terms, follow these steps:
  • List the factors: Break down each term into its prime factors or basic components.
  • Identify common factors: Find the factors that appear in both lists.
  • Select the greatest common factor: Choose the highest factor common to both terms.

In the given example, \(x^{3} - 5x\), the GCF is \(x\).
This is because \(x\) is the shared factor in both \(x^{3}\) and \(-5x\).
Factoring out the GCF simplifies the expression and makes further algebraic manipulation easier.
Polynomial Factoring
Factoring polynomials involves breaking down a polynomial into simpler components (or factors) that, when multiplied together, give the original polynomial.
Factoring is a critical skill for solving polynomial equations and simplifying expressions.

To factor a polynomial, follow these general steps:
  • Find the GCF: Identify and factor out the greatest common factor from all terms.
  • Apply factoring techniques: Depending on the polynomial, different techniques like grouping, using formulas, or special patterns (like the difference of squares) may apply.

In the exercise \(x^{3} - 5x\), factoring begins by identifying the GCF, which is \(x\).
Factoring out \(x\) from each term gives \(x(x^{2} - 5)\).
This simplifies the polynomial and sometimes reveals further opportunities for factoring or solving.
Practice different types of factoring to become proficient, as different polynomials may require different strategies.

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Most popular questions from this chapter

Factor each polynomial by grouping. $$ x^{2} a^{2}+a^{2}+x^{2}+1 $$

Factor each polynomial by grouping. $$ a x-2 a+4 x-8 $$

Factor each polynomial using the trial-and-error method. $$ a^{2}+15 a+54 $$

Firing a slingshot. A girl uses a slingshot to propel a stone upward at 64 feet per second from a window that is 80 feet above the ground. The height of the stone above the ground (in feet) at time \(t\) (in seconds) is given by $$h(t)=-16 t^{2}+64 t+80$$ Find the time at which the stone strikes the ground.

Solve each equation. $$16 x^{2}+8 x+1=0$$
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